Robust adaptive fuzzy controller applied to double inverted pendulum

Abstract

The main objective of the present work is to stabilize and maintain the angular position of Double Inverted Pendulum (DIP) system at desired position in presence of disturbances and noise. The system is highly coupled, nonlinear, complex and unstable, thereby making it difficult to control. Genetic algorithm tuned Fuzzy Controller (GFC) and adaptive Neuro-Fuzzy Controller (NFC) is proposed for the purpose, wherein the fuzzy parameters are optimized by genetic algorithm and artificial neural network respectively. The adaptive neuro-fuzzy control technique enjoys powerful learning capability of neural network, whereas genetic algorithm discovers the optimum solutions for the problem. Also a suitable function is proposed for modifying training data set of neuro-fuzzy inference system that leads to Modified Neuro-Fuzzy Controller (MNFC). Linear Quadratic Regulator (LQR) and Fuzzy Logic Controllers (FLC) are also designed for comparative analysis. Intensive simulation studies are carried out to critically examine the performance of designed controllers on the basis of Integral Absolute Error (IAE), settling time, overshoot and steady state error for set-point tracking, disturbance rejection, noise suppression and simultaneous noise & disturbance rejection. The rigorous comparative analysis shows that MNFC exhibits fast and robust control of DIP system in comparison to designed controllers for all cases.

Keywords

Double Inverted Pendulum (DIP)Fuzzy Logic Controller (FLC)Genetic Algorithm (GA)Linear Quadratic Regulator (LQR)GA Tuned Fuzzy Controller (GFC)Modified Neuro-Fuzzy Controller (MNFC)

1 Introduction

Double inverted pendulum system is an ideal platform for control engineers to apply and validate various control techniques because of its high nonlinearity, open loop instability and multi-dimensionality [1]. The system becomes more complex and difficult to control due to large number of variables and highly coupled states. In order to stabilize the system a suitable control force needs to be applied on cart so that pendulums remain at the desired upright position. In recent years, combinations of modern control techniques with intelligent algorithms are employed to control such systems. These hybrid controllers exhibit the advantages of both the techniques which lead to optimal control action.

LQR being a modern control technique is applied successfully for control of various systems. Zhang and Zhang [2] designed a self-adjusting LQR controller for planar inverted pendulum, in which output is reformed by optimizing factor which adjusts the control action according to state variables. Results ensure fast response, better stability and robustness in different working environments. An optimal LQR controller [3] with well-tuned weighting matrices for self-erecting single inverted pendulum is used for stabilization in vertical position. The proposed LQR controller makes stabilization process faster and smoother with less oscillations and better robustness than a full state feedback controller. An alternative control scheme based on finite element method is developed by Dhang et al. for optimal control ofsingle and double inverted pendulum [4]. Zilicet al. [5] designed the state feedback LQ and LQG controller for pneumatically-actuated inverted pendulum. Simulation results prove the effectiveness of designed controllers.

In last few decades, researchers have focused on the intelligent techniques such as fuzzy logic, neural networks etc. This leads to advancement of intelligent human-like control of complex systems. Literature [6 –11] reveals the use of fuzzy logic reasoning for design of fuzzy controllers for various applications. The main problem of fuzzy inference system is that, if all the input variables are considered as antecedent part of fuzzy system, then number of fuzzy rules increase exponentially with increase of input variables which leads to poor ability in expressing the control priority orders. To deal with the problem of rule explosion Yi and Yubazaki designed [12] a fuzzy controller for stabilization of a pendulum on cart, based on Single Input Rule Module (SIRM) dynamically connected fuzzy inference model. SIRM and dynamic importance degree are also successfully implemented for control of series and parallel type of double inverted pendulum on cart. The priority orders of the cart and angles are automatically adjusted according to control situations with pendulum angle having higher priority [13, 14]. Fuyan Cheng et al. designed a fuzzy controller [15] using optimal control theory for stabilizing a real time double inverted pendulum system. A composition coefficient for the state variables is evaluated with the help of computer-aided design (CAD), to reduce the number of variables and simplify the design of fuzzy logic controller. The same multivariate problem is also solved using sensor data fusion technique with fuzzy logic control to balance wine glass on inverted pendulum [16].

Being an intelligent technique fuzzy logic proves to be efficient control methodology, but the problem lies in intuition based selection of parameters related to membership functions and rule base, which requires expertise. In order to overcome this problem meta-heuristic algorithms or neural networks are incorporated to fuzzy system [17 –23]. Various optimization techniques namely Genetic Algorithm (GA), Ant Colony Optimization (ACO), cuckoo Search, Particle Swarm Optimization (PSO), Differential Evolution (DE) etc. are discussed in literature [24 –27]. In this paper, genetic algorithm is utilized as it follows natural selection process without requiring any problem-specific information, search space and also operates satisfactorily on search spaces having gaps, jumps, or noise. It searches in parallel, with numerous points on the problem space in different directions [28]. A genetic algorithm based tuning of fuzzy control rule base is proposed by Herrera et al. [17] so that fuzzy controller behaves as an expert operator. Results obtained prove the good performance of methodology. Genetic algorithm is employed to optimize the control rules of the fuzzy system by Varsek et al. First the rules are automatically changed into understandable form by means of inductive machine learning and then genetic algorithm is used to optimize the numerical parameters of the induced rules [18].

It is discussed in literature that learning capability of artificial neural network is incorporated with fuzzy system to overcome the aforementioned issues [21 –23]. Jang [23] presented ANFIS (Adaptive-Network-based Fuzzy Inference System), for mapping of input-output data set using a hybrid learning method. Simulation results of ANFIS, Artificial Neural Network (ANN) and fuzzy logic are compared for different nonlinear functions. ANFIS performed better as compared to other techniques. The self-erecting single inverted pendulum is controlled using two control loops i.e. swing up control and stabilization. A position-velocity controller is used to swing up from stable state to unstable state and an ANFIS controller is designed to maintain stability at unstable point. Simulation results show the efficacy of the control policy [29]. Adaptive neuro fuzzy technique is successfully implemented in different fields of engineering such as wind turbine system [30], dc motor [31, 32] etc. Numerous adaptive control techniques for stabilization of inverted pendulum have been reported in literature. Rudra and Barai [33] designed a robust adaptive back-stepping control law for inverted pendulum. The designed controller is tested for tracking performance and parameter convergence rate of the system. Simulation results prove the effectiveness of the designed controller. Wang [34] proposed a nonlinear adaptive fuzzy sliding mode controller for the position tracking of inverted pendulum system. Simulation studies show the usefulness of the proposed control law.

The objective of the present work is to balance pendulums in upright position by application of a suitable control force on the cart in presence of actuator disturbances and sensor noise. The disturbance arising in DIP system is due to cogging and eccentricity, occurring because of the tendency of rotor to be locked at specific points. Thus actuator experiences sinusoidal disturbance forces [35]. Further noise is generated within the sensor due to aging, environmental conditions, inherent shortcomings of sensor etc. In order to deal with disturbance and noise generated within system, a fuzzy logic controller is designed and to improve its performance, parameters associated with membership functions and rules are optimized using genetic algorithm and neural network. Thus GA-fuzzy controller and adaptive neuro-fuzzy controller are designed for control of DIP system. Further a modified neuro fuzzy controller is proposed with suitable changes in manipulating variable of the training data, which improves the transient as well as steady state performance of the system. The conventional LQR and fuzzy logic controllers are also designed for comparative study. The robustness of designed controllers is tested for set-point tracking, disturbance rejection, noise suppression and simultaneous noise as well as disturbance rejection. A quantitative comparison on the basis of IAE value is carried out to critically evaluate the performance of controllers.

The paper is organized as follows: The model of double inverted pendulum system is discussed in the Section 2, Section 3 discusses the design of controllers. The results and discussion are presented in the Sections 4 and 5 concludes the research work.

2 Description of double inverted pendulum

The double inverted pendulum system consists of a cart on a rail, two pendulums and encoders. The two pendulums are joined serially and the starting point of first pendulum is attached to the cart. Encoders are used to measure the position and velocity of cart and pendulums. Figure 1 shows double inverted pendulum on cart [15] and values of various parameters of the system are given in Table 1. The behavior of the system is dynamic because of continuous variation in velocity and position of pendulums in response to force applied on cart. The inter-dependency of velocity, position and force of the system is derived by use of a set of differential equations known as Euler Lagrange equations of motion.

Fig.1

Schematic diagram of a Double Inverted Pendulum system.

Table 1

Parameters of DIP system

m₀, m₁, m₂, m₃	Mass of the cart, first pole, second pole, joint 5.75 Kg, 1.52 Kg, 0.51 Kg, 0.75 Kg
L₁, L₂	Length of first pendulum (= 2l₁) and length of second pendulum (= 2l₂) 1 m, 1.5 m
g	Acceleration due to gravity, 9.8 m/s²
θ₁, θ₂	The angle between pole 1, 2 with vertical direction in radian
u	Applied force on cart (N)

The Euler Lagrange formulation is used to obtain the dynamic equations as $\frac{d}{dt} \frac{dH}{d {\dot{q}}_{i}} - \frac{dH}{{dq}_{i}} = F_{i}$ (1) where, H = G - V is a Lagrangian, F_i is a force vector applied on the generalized coordinates q_i, V and G are potential and kinetic energy of the system respectively. These energies are the sum of the energy of cart and both pendulums, which is derived from basic laws of physics as: $\begin{matrix} G & = & \frac{1}{2} (m_{0} + m_{1} + m_{2} + m_{3}) {\dot{x}}^{2} \\ + (\frac{2}{3} m_{1} l_{1}^{2} + 2 m_{2} l_{1}^{2} + 2 m_{3} l_{1}^{2}) {\dot{θ}}_{1}^{2} + \frac{2}{3} m_{2} l_{2}^{2} {\dot{θ}}_{2}^{2} \\ + (m_{1} l_{1} + 2 m_{2} l_{1} + 2 m_{3} l_{1}) \dot{x} {\dot{θ}}_{1} cos θ_{1} \\ + m_{2} l_{2} \dot{x} {\dot{θ}}_{2} cos θ_{2} + 2 m_{2} l_{1} l_{2} cos (θ_{1} - θ_{2}) {\dot{θ}}_{1} {\dot{θ}}_{2} \end{matrix}$ (2) $\begin{matrix} V & = & {gl}_{1} cos θ_{1} + 2 m_{3} {gl}_{1} cos θ_{1} \\ + m_{2} g (2 l_{1} cos θ_{1} + l_{2} cos θ_{2}) \end{matrix}$ (3)

Therefore, Lagrangian H is

$\begin{matrix} H & = & \frac{1}{2} (m_{0} + m_{1} + m_{2} + m_{3}) {\dot{x}}^{2} \\ + (\frac{2}{3} m_{1} l_{1}^{2} + 2 m_{2} l_{1}^{2} + 2 m_{3} l_{1}^{2}) {\dot{θ}}_{1}^{2} + \frac{2}{3} m_{2} l_{2}^{2} {\dot{θ}}_{2}^{2} \\ + (m_{1} l_{1} + 2 m_{2} l_{1} + 2 m_{3} l_{1}) \dot{x} {\dot{θ}}_{1} cos θ_{1} \\ + m_{2} l_{2} \dot{x} {\dot{θ}}_{2} cos θ_{2} + 2 m_{2} l_{l} l_{2} cos (θ_{1} - θ_{2}) {\dot{θ}}_{1} {\dot{θ}}_{2} \\ - m_{1} {gl}_{1} cos θ_{1} - 2 m_{3} {gl}_{1} cos θ_{1} \\ - m_{2} g (2 l_{1} cos θ_{1} + l_{2} cos θ_{2}) \end{matrix}$ (4)

Substituting the value of Lagrangian H in Equation (1) and replacing the generalized coordinate with θ₁ and θ₂ yields $\frac{d}{dt} \frac{\partial H}{\partial {\dot{θ}}_{1}} - \frac{\partial H}{\partial θ_{1}} = 0$ (5) $\frac{d}{dt} \frac{\partial H}{\partial {\dot{θ}}_{2}} - \frac{\partial H}{\partial θ_{2}} = 0$ (6)

Or explicitly: $\begin{matrix} (\frac{4}{3} m_{1} l_{1}^{2} + 4 m_{2} l_{1}^{2} + 4 m_{3} l_{1}^{2}) {\ddot{θ}}_{1} \\ + (m_{1} l_{1} + 2 m_{2} l_{1} + 2 m_{3} l_{1}) \\ \ddot{x} cos θ_{1} + 2 m_{2} l_{l} l_{2} {\ddot{θ}}_{2} cos (θ_{1} - θ_{2}) \\ + 2 m_{2} l_{1} l_{2} {\dot{θ}}_{2}^{2} sin (θ_{1} - θ_{2}) \\ - (m_{1} l_{1} + 2 m_{2} l_{1} + 2 m_{3} l_{1}) g sin θ_{1} = 0 \end{matrix}$ (7) $\begin{matrix} m_{2} l_{2} \ddot{x} cos θ_{2} + 2 m_{2} l_{1} l_{2} {\ddot{θ}}_{1} cos (θ_{1} - θ_{2}) \\ + \frac{4}{3} m_{2} l_{2}^{2} {\ddot{θ}}_{2} - 2 m_{2} l_{1} l_{2} {\dot{θ}}_{1}^{2} sin (θ_{1} - θ_{2}) \\ - m_{2} l_{2} gsin θ_{2} = 0 \end{matrix}$ (8)

The operating point of the system is at angles (θ₁, θ₂) = (0, 0) i.e. when both the pendulum angles are zero. A small perturbation is introduced around operating point and expanded using Taylor series. Therefore cos(θ₁ - θ₂) ≅1, sin(θ₁ - θ₂) ≅ θ₁ - θ₂, cos θ₁ ≅ cos θ₂ ≅ 1, sin θ₁ ≅ θ₁, sin θ₂ ≅ θ₂. Thus linearization is done at unstable operating point and linear time invariant state model of the system is obtained from Equations (7 and 8) as follows.

$\begin{matrix} [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 & a_{4} & a_{3} & a_{5} \end{matrix}] [\begin{matrix} \dot{x} \\ {\dot{θ}}_{1} \\ {\dot{θ}}_{2} \\ \ddot{x} \\ {\ddot{θ}}_{1} \\ {\ddot{θ}}_{2} \end{matrix}] \\ = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & f_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & f_{2} & 0 & 0 & 0 \end{matrix}] [\begin{matrix} x \\ θ_{1} \\ θ_{2} \\ \dot{x} \\ {\dot{θ}}_{1} \\ {\dot{θ}}_{2} \end{matrix}] + [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix}] \\ Let a = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 & a_{4} & a_{3} & a_{5} \end{matrix}], \\ b = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & f_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & f_{2} & 0 & 0 & 0 \end{matrix}] and c = [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix}] \\ \begin{matrix} [\begin{matrix} \dot{x} \\ {\dot{θ}}_{1} \\ {\dot{θ}}_{2} \\ \ddot{x} \\ {\ddot{θ}}_{1} \\ {\ddot{θ}}_{2} \end{matrix}] = A [\begin{matrix} x \\ θ_{1} \\ θ_{2} \\ \dot{x} \\ {\dot{θ}}_{1} \\ {\dot{θ}}_{2} \end{matrix}] + B * u (t) \\ y (t) = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{matrix}] [\begin{matrix} x \\ θ_{1} \\ θ_{2} \\ \dot{x} \\ {\dot{θ}}_{1} \\ {\dot{θ}}_{2} \end{matrix}] + [\begin{matrix} 0 \\ 0 \end{matrix}] u (t) \end{matrix}} \end{matrix}$ (9)

Where $\begin{matrix} A & = & a^{- 1} b, B = a^{- 1} c \\ a_{1} & = & m_{1} l_{1} + 2 m_{2} l_{1} + 2 m_{3} l_{1} \\ a_{2} & = & \frac{4}{3} m_{1} l_{1}^{2} + 4 m_{2} l_{1}^{2} + 4 m_{3} l_{1}^{2} \\ a_{3} & = & 2 m_{2} l_{1} l_{2}, a_{4} = m_{2} l_{2} \\ a_{5} & = & \frac{4}{3} m_{2} l_{2}^{2} \\ f_{1} & = & (m_{1} l_{1} + 2 m_{2} l_{1} + 2 m_{3} l_{1}) \\ and f_{2} = m_{2} l_{2} g \end{matrix}$

The model of DIP system is simulated by using Equation (9) and force ‘u’ applied to the cart is the manipulating variable obtained from controller. The displacement of cart position ‘x’ is restricted in the range –3 to 3 meters. The state variable model is used to analyze the stability, controllability and observability of the system. The roots of the characteristic equation are obtained and it is observed that two roots are lying on the right side of s-plane and therefore the system is unstable. Thus in order to stabilize the system there is a need to design a suitable controller. In the present work state feedback controller is designed for the purpose. The rank of composite matrix is found to be same as that of the order of the system, thus the system is completely controllable and observable in accordance with Kalman’s test. As the system is controllable its Eigen values can be arbitrarily located by complete state feedback control technique.

3 Design of controllers

The main objective is to stabilize and maintain the angular position of two pendulums at zero radians under the effect of sensor noise and actuator disturbance. This is achieved by applying a suitable horizontal force to the cart. The controllers designed for controlling the system are discussed below.

3.1 Linear quadratic regulator

A linear quadratic regulator is designed by minimizing the performance index J. $J = \int_{0}^{\infty} (X^{T} DX + u^{T} Su) dt$ (10) where $X = {[x, θ_{1}, θ_{2}, \dot{x}, {\dot{θ}}_{1}, {\dot{θ}}_{2}]}^{T}$ is the state vector comprising of state variables, D is a positive-semi definite matrix and S is a positive-definite matrix. The force u is determined, by solving algebraic Riccati equation given below $A^{T} M + MA - {MBS}^{- 1} B^{T} M + D = 0$ (11) where M is the Riccati matrix. Hence optimal control input u for any initial state X (0) is given by: $u (t) = - K * X = - S^{- 1} B^{T} M * X$ (12) where K is optimal feedback gain matrix, D and S are chosen as $D = diagonal ([100 200 300 0 0 0]) and S = 1$ (13)

Therefore, $K = - S^{- 1} B^{T} M = [k_{1} k_{2} k_{3} k_{4} k_{5} k_{6}]$ (14)

On the basis of Equation (10 to 14) LQR controller is designed and control input ‘u’ for the system is obtained. The system is highly unstable and dynamic, therefore intelligent controllers are proposed for the control purpose. The fuzzy logic controller is explained in the next section.

3.2 LQR based fuzzy logic controller

Fig.2

Fuzzy logic control structure.

Fuzzy logic provides a simple way to control a system, without the use of actual mathematical model of the system. The uncertainty and instability of double inverted pendulum system makes fuzzy controller a good choice of control policy. Fuzzy control structure developed is shown in Fig. 2 which consists of DIP system, fusion function, scaling gains and fuzzy logic controller. As discussed above the DIP system is a multivariate system having six state variables. Due to high dimensionality of the system there is a problem of rule explosion in designing fuzzy logic controller. If five membership functions are assigned to each of the six state variables, the number of rules needed is 5⁶ (15625). Constructing such a rule base is a tedious task for designer. It is revealed from the literature [15 , 36] that, the problem may be solved using a sensory fusion designed by state feedback gain of LQR, which reduces the dimension for fuzzy controller. $F (x) = [\begin{matrix} \frac{k_{1}}{k_{3}} & \frac{k_{2}}{k_{3}} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{k_{4}}{k_{6}} & \frac{k_{5}}{k_{6}} & 1 \end{matrix}]$ (15) $[\begin{matrix} E \\ EC \end{matrix}] = F (x) X$ (16) $E = \frac{k_{1}}{k_{3}} x + \frac{k_{2}}{k_{3}} θ_{1} + θ_{2}$ (17) $EC = \frac{k_{4}}{k_{6}} \dot{x} + \frac{k_{5}}{k_{6}} {\dot{θ}}_{1} + {\dot{θ}}_{2}$ (18)

Fusion function ‘F (x)’ is chosen in such a way that all the non-derivative states are fused to form error ‘E’ and the derivative states are fused to form rate of change of error ‘EC’ as given in Equations (17 and 18) respectively. Thus the dimension of the system is reduced to two i.e. comprehensive error E and the rate of change of error EC, which impressively simplifies the design of fuzzy logic controller. The two fused states are scaled by gains K_E = 17 and K_EC = 6.2, and applied as input to the fuzzy logic controller. The values of these gains are obtained by trial and error. Input variables E and EC are fuzzified by Gaussian membership functions. The selection of membership function depends on the internal behavior of the system. In this work, five Gaussian membership functions are used for both inputs and output which are labeled as NM (negative medium), NS (negative small), ZE (zero), PS (positive small), and PM (positive medium).

The error E and rate of change of error EC are inputs to the fuzzy logic controller and called as antecedent or premise. U is the control output and called as consequence or conclusion. The conclusion is evaluated by the fuzzy logic controller so as to reduce the error and rate of change of error between the actual and desired position of the system. Further the rule base (Table 2) for FLC is developed on the basis of knowledge and experience gained by analyzing the dynamics of system. A fuzzy If-Then rule is expressed as: $If E is NM and EC is NS then U is NM .$

Table 2

Rule base for FLC

E	EC	NM	NS	ZE	PS	PM
	U
NM	NM	NM	NS	NS	ZE
NS	NM	NS	NS	ZE	PS
ZE	NM	NS	ZE	PS	PM
PS	NS	ZE	PS	PS	PM
PM	ZE	PS	PS	PM	PM

The inputs E and EC are mapped to suitable membership functions. The inference system invokes each appropriate rule on the basis of inputs and generates a fuzzified output. The Mamdani inference technique is used to infer the output. The fuzzified output is converted to crisp value with the help of centroid defuzzification method, which returns the centre of area under the curve as crisp output. 3-dimensional surface showing variation of output U with respect to E and EC is shown in Fig. 3. Gain K_U = 25, obtained by trial and error, is used for scaling of output U and then applied to the cart of system.

Fig.3

Surface view of Fuzzy logic control.

The main drawback of fuzzy logic controller is the intuition based selection of gains and parameters related to the membership functions. The problem is solved by a suitable optimization technique. In the present case, genetic algorithm is used, which emends the span of inputs and adapts the membership functions accordingly to realize optimal fuzzy logic controller. The fuzzy logic controller optimized with the help of GA leads to a GA-fuzzy logic controller as explained in next section.

3.3 GA-Fuzzy logic controller

Fig.4

GA-fuzzy control structure.

As discussed previously the optimal selection of gains and parameters associated with membership function is required for efficient fuzzy logic controller. Generally the span and parameters of membership functions are selected intuitively to design FLC. In order to design the efficient fuzzy logic controller genetic algorithm is used for optimization of parameters associated with membership functions and gains (K_U, K_E and K_EC), which leads to a Genetic Algorithm tuned Fuzzy Logic Controller (GFC). Figure 4 shows the schematic diagram of GFC. The conventional FLC designed previously is a two-dimensional fuzzy controller and has 5 + 5 + 5 = 15 membership functions. Each membership function has two parameters i.e. centre and width {γ_i, σ_i} and span of both inputs and output needs to be optimized. Thus total number of parameters to be optimized are 39 (i.e. parameters of membership functions 15*2 = 30, span of inputs and output membership function is 2 + 2 +2 = 6 and scaling gain of inputs and output of controller 1 + 1 +1 = 3). The lower and upper bounds of each variable are defined based on FLC designed earlier. The multi objective fitness function for optimizing the fuzzy controller is weighted sum of overshoot and settling time of both the pendulums and cart position as given in Equation (19). $\begin{matrix} Z & = & α * ({St}_{x} + {St}_{θ_{1}} + {St}_{θ_{2}}) \\ + β * ({Ov}_{x} + {Ov}_{θ_{1}} + {Ov}_{θ_{2}}) \end{matrix}$ (19) $α = 0.4 and β = 0.6$ where St stands for settling time and Ov is overshoot. The values of α and β are selected so that more weightage is given to minimization of overshoot as compared to settling time.

Genetic algorithm is a random search method that copies the method of biological evolution. GA begins without prior information of the exact solution. It depends solely on responses from its surroundings and evolution operators (i.e. reproduction, crossover and mutation) to get better solution. It starts from various independent points and explores them in parallel. These capabilities improve the performance of GA for complex fields with high dimensionality. The typical start of the GA process is with a random population that is formed by a real-valued number or a binary string called chromosome. The fitness of each chromosome is evaluated and survival of the fittest scheme is applied. The algorithm is described briefly with the sequence of steps as follows:

Generate random population having 50 sets of chromosomes for 39 variables of FLC, on the basis of upper and lower bound of variables.

Evaluate the fitness function value Z for every chromosome in population by simulating DIP system with FLC controller.

Generate new population using the subsequent steps:

Select parent chromosomes from a population according to their fitness value i.e. lower the fitness value, higher is the chance to be selected.

Crossover is performed on the parent chromosomes to form new offspring.

To increase the diversity in population, mutation operation is performed on the offspring.

Place new offspring’s in the new population.

If the stopping criterion is met i.e. function tolerance is 10^-6 or maximum number of iterations (100) is achieved, then stop and return the best solution from current population.

Otherwise, go to step 2.

50 chromosomes are generated for each variable in a single iteration. Rank method is used for fitness scaling in which the raw scores are scaled on the basis of rank of each chromosome, rather than its score. Therefore rank of the fittest chromosome is 1, the next fittest is 2, and so on. Then tournament selection function is used for choosing parents for the succeeding generation based on their scaled values. Now parents are crossover using scattered method, to form new chromosomes for the next generation. After crossover, to increase the diversity in the search space, adaptive feasible mutation operation is performed in which flipping of a bit is done. The fitness of newly generated chromosome is evaluated online and the process is repeated until the stopping criterion of function tolerance is met or last iteration is performed. Thus parameters of membership function are adapted and the optimized fuzzy controller is realized. GA converges to the optimal solution in 64 iterations with the fitness value Z = 8.6 as shown in Fig. 5(a). Values obtained for gains are K_E = 26.361, K_EC = 9.474, K_U = 31.419. The premise parameters for membership functions of input E and EC along with the conclusion parameters of output membership function U for GFC controller are given in Table 3. The optimized membership functions for two inputs, output and control surface of the GA-fuzzy logic control are shown in Fig. 5. It is observed from Fig. 5(b), (c) and (d) that shape of membership functions are optimized and covers the entire span of the inputs and output. The wider membership functions in inputs have dominant effect in selecting the output membership function and vice versa.

Fig.5

Optimized GA fuzzy controller (a) convergence curve (b) Membership functions of E(c) Membership functions of EC (d) Membership functions of U and (e) Control surface.

Table 3

Parameters associated with membership function of GFC controller

Membership functions	Premise Parameters				Conclusion Parameters
	E		EC		U
	γ ₁	σ ₁	γ ₂	σ ₂	γ ₃	σ ₃
NM	–3.43	0.849	–4.971	0.4537	–7.502	2.332
NS	–1.076	1.454	–2.097	0.5459	–2.6	1.314
ZE	0.721	3.87	–0.856	0.47	–1.695	3.161
PS	2.176	2.038	2.206	1.838	2.957	1.814
PM	4.94	0.223	5.87	2.06	7.72	3.96
Span	[–3.245, 4.97]		[–5.566, 6.618]		[–7.382, 7.943]

The parameters of fuzzy logic controller may also be optimized with the help of artificial neural network. Such structure is referred as adaptive neuro-fuzzy inference system. The technique is also tested for the control of DIP system and is discussedbelow.

3.4 Neuro-fuzzy controller (NFC)

Adaptive neuro-fuzzy inference technique is an approach of transforming the burden of designing fuzzy logic control and decision systems to the training and learning of connectionist neural networks. First-order Takagi–Sugeno fuzzy model is used to design the NFC. Control structure for NFC controller is shown in Fig. 6. It is depicted in the block diagram that the parameters of the fuzzy logic controller are optimized with the help of neural network learning. It has two inputs, error (e) and rate of change of error ( $\dot{e}$ ) and the output is control signal u. The ith IF–THEN rule is given as: $\begin{matrix} If e is A_{j} and \dot{e} is B_{j}; then \\ u_{i} = P_{i} * e + Q_{i} * \dot{e} + R_{i} \end{matrix}$ and $\begin{matrix} [\begin{matrix} e \\ \dot{e} \end{matrix}] & = & [z_{c}] * X \\ Z_{c} & = & [\begin{matrix} k_{1} & k_{2} & k_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & k_{4} & k_{5} & k_{6} \end{matrix}] \end{matrix}$ where A_j and B_j are the sets of fuzzy membership assigned to input variable e and $\dot{e}$ , j = 1, 2 … s and i = 1, 2 … s², s is the number of membership functions taken for the inputs e and $\dot{e}$ , u_i is the first order linear output function expressed as weighted sum of inputs, P_i, Q_i and R_i are output parameters of NFC and k₁, k₂, k₃, k₄, k₅ and k₆ are the gains of LQR controller. The 5 layer NFC architecture with two inputs and one output is shown in Fig. 7.

Fig.6

Block diagram of neuro-fuzzy controller.

Fig.7

Five layered NFC architecture.

In layer 1, every node is an adaptive node, with individual membership function specifying the degree of inputs to satisfy the quantifier. A_j and B_j are considered to be Gaussian membership functions and their representations are given in Equations (20 and 21). $\begin{matrix} L_{1, j} & = & μ A_{j} (e, γ, σ) \\ = & \exp^{\frac{- {(e - γ)}^{2}}{2 σ^{2}}} j = 1, 2, 3 \dots s \end{matrix}$ (20) $\begin{matrix} L_{1, j} & = & μ B_{j} (\dot{e}, γ, σ) \\ = & \exp^{\frac{- {(\dot{e} - γ)}^{2}}{2 σ^{2}}}, j = 1, 2, 3 \dots s \end{matrix}$ (21)

The parameters for fuzzy membership functions are central value ‘γ’ and standard deviation σ > 0. These parameters are known as premise parameters. The change in these parameters due to the data set considered for training, adjusts the shape of Gaussian membership function.

In layer 2, every node is fixed and denoted by Π, output is the product of all incoming signals and is given in Equation (22).

$\begin{matrix} L_{2, i} & = & W_{i} = [\begin{matrix} W_{1} & \dots & W_{s} \\ ⋮ & ⋱ & ⋮ \\ W_{s^{2} - (s - 1)} & \dots & W_{s^{2}} \end{matrix}] \\ = & [\begin{matrix} μ A_{1} * μ B_{1} & \dots & μ A_{1} * μ B_{s} \\ ⋮ & ⋱ & ⋮ \\ μ A_{s} * μ B_{1} & \dots & μ A_{s} * μ B_{s} \end{matrix}] \end{matrix}$ (22)

Every node of the second layer represents the firing strength of the associated rule. The firing strength (W_i), is obtained by T-norm operator which is algebraic product (A_j × B_j).

In layer 3, every node is a static node denoted as N and output of ith node is the ratio of firing strength of the ith rule (W_i) to the summation of the firing strength of all the rules, as given in Equation (23). $L_{3, i} = {\bar{W}}_{i} = \frac{W_{i}}{\sum_{i = 1}^{s^{2}} W_{i}}$ (23)

Every node in layer 4 is an adaptive node with a node function given by the Equation (24). $L_{4, i} = {\bar{W}}_{i} * (P_{i} * e + Q_{i} * \dot{e} + R_{i})$ (24)

Where P_i, Q_i and R_i are the sets of control variables and ${\bar{W}}_{i}$ is known as normalized firing strength generated from layer 3. Parameters in this layer are defined as output parameters. Layer 5 is designated by Σ which represents a single fixed node. It calculates the global output as the sum of all entering signals and is given by the Equation (25).

$\begin{matrix} u_{i} & = & L_{5} = \sum_{i} {\bar{W}}_{i} * (P_{i} * e + Q_{i} * \dot{e} + R_{i}) \\ = & \frac{\sum_{i} W_{i} * (P_{i} * e + Q_{i} * \dot{e} + R_{i})}{\sum_{i} W_{i}} \end{matrix}$ (25)

A hybrid learning algorithm is utilized to identify NFC parameters. Initially, input membership functions and number of rules for fuzzy inference system are framed according to the input–output training data sets. These membership functions can be assigned for input variables, either by plotting the data sets and examining them visually or simply by trial and error approach. But these methods are not very effective and to overcome the disadvantages, clustering methods are utilized. Therefore initial membership function and number of fuzzy rules are generated using grid partition clustering method. In this method, the number of membership functions on each input variable uniquely determines the number of rules. There are two inputs and five membership functions of each input which results in 5^∧2 = 25 fuzzy IF–THEN rules. The Gradient descent and least square estimation learning algorithms are combined for fast identification of premise and consequent parameters of NFC.

The training data set is generated by simulating the system with LQR controller using 4th order Runge-Kutta differential equation solver having the sampling time of 0.01 sec. Initial state is considered as (0.25 0.120.10 0 0) ^T and a noise of 0.1 radian is incorporated in 2nd pendulum for time interval of 4 to 4.1 s. Total 1201 samples of input-output data set are obtained in 12 s. The error e, rate of change of error $\dot{e}$ and the corresponding force on the cart u are obtained under these conditions. It is observed from the simulation results (Fig. 8a) that there are large transients in the response due to noise, which implies that LQR controller does not efficiently control the noisy system. In order to minimize the transients and improve the steady state performance of the system a necessary modification in the training data set is proposed. The modification is made by subtracting a function h₁ (y) from the manipulating variable. Where h₁ (y) is selected in such a way that its effect is more during transients. The samples at steady state are modified by a multiplying function h₂ (y), in order to drive the system output towards steady state rapidly. The following modification functions given in Equations (26 and 27) are found suitable for the required purpose and are achieved by rigorous analysis of data. Modified training data u_m is shown in Fig. 8(b). $h_{1} (y) = y * \frac{1}{548}, 1 \leq y < 548$ (26) $h_{2} (y) = y * \frac{1}{1201}, 548 \leq y \leq 1201$ (27) $u_{m} (y) = {\begin{matrix} u (y) - h_{1} (y), & 1 \leq y < 548 \\ u (y) * h_{2} (y), & 548 \leq y \leq 1201 \end{matrix}$ (28)

Where y is an integer and u (y) * h₂ (y) is calculated as

$\begin{matrix} u (y) * h_{2} (y) \\ = {[diagonal (u (y))]}_{654 \times 654} \\ * {[\frac{y}{1201}]}_{654 \times 1} for 548 \leq y \leq 1201 \end{matrix}$ (29)

The modified data so obtained is used to train the MNFC controller offline. Fuzzy inference system is generated using grid partition clustering method with two inputs and one output. Inputs are fuzzified with 5 Gaussian membership functions and first order output is employed. Finally the parameters of fuzzy inference system are optimized using hybrid training algorithm, with epochs as 60 and tolerance for error as zero. Error obtained after training is 1.37626 and the error plot is shown in Fig. 9(a). The structure of MNFC with twenty five rules is shown in Fig. 9(b). The surface view of the fuzzy system without training and after training is shown in Fig. 10(a) and (b) respectively.

Fig.8

(a) Pendulum angles controlled by LQR controller and (b) Modified training data.

Fig.9

(a) Training error (b) MNFC structure.

Fig.10

Control surface of MNFC (a) Initial surface view (b) Final surface view after training.

The results obtained from MNFC are compared with NFC as shown in Fig. 11. It is observed from the results that modification in the manipulating variable improves the performance of the system significantly. The integral absolute error (IAE) of the two pendulum angles is obtained to be 0.168 and 0.1013 for MNFC whereas it is 0.216 and 0.1296 for NFC. This is due to reason that the suggested modification attempts to adjust the manipulating variable during transients as well as steady state conditions.

Fig.11

Comparison of NFC and MNFC response (a) 1st pendulum angle (b) 2nd pendulum angle.

4 Results and discussion

The robustness of designed controllers for the control of double inverted pendulum is investigated by conducting intensive simulation studies. The designed controllers are rigorously examined for set-point tracking, disturbance rejection, noise suppression and simultaneous noise and disturbance rejection. All the simulations are performed in MATLAB on Intel^®, Core™i3, 4GB RAM, 1.70 GHz personal computer. The algebraic solver used for the simulation is Runge-kutta with sampling time of 0.001 s. Further, the performance and robustness of designed controllers are compared on the basis of integral absolute error (IAE), settling time, overshoot and steady state error of both the pendulum angles.

4.1 Set-point tracking performance

As discussed previously, the control input u in Equation (12) is obtained using the matrix K which in turn is obtained by minimizing the performance index JEquation (10). Thus LQR controller is designed for control and stabilizing of pendulum angles by generating control input u. It is observed from the results (Fig. 12) that the pendulum angles obtained are oscillatory, have large overshoots and steady state is reached at a later stage. Therefore, in order to improve the control performance fuzzy logic controller is designed. Fusion function is incorporated to reduce the dimension of input variables to error E and rate of change of error EC which greatly simplifies the design of FLC. The performance of designed FLC is slightly better than LQR controller. The integral absolute error, oscillations and settling time are reduced as compared to LQR. The membership functions and the rules associated with fuzzy logic control are selected on the basis of expert’s intuition.

Fig.12

Set-point tracking performance of MNFC, GFC, FLC and LQR (a) 1st pendulum angle (b) 2nd pendulum angle (c) Controller output.

The performance of the designed FLC may further be improved if the parameters of FLC are tuned using an optimization technique. In the present case genetic algorithm is considered for the purpose which leads to GFC. GA searches the parameters of fuzzy in unrestricted and multidirectional manner to produce optimized fuzzy inference system. Response of the system with GFC is much better than FLC and LQR controller. Further, neural network learning capability is utilized for adapting the parameters of TS fuzzy system. The network is trained by modified data obtained from LQR controller, which leads to modified neuro-fuzzy controller (MNFC). The set-point tracking performance comparison is shown in Fig. 12. It is obvious from the figure that MNFC controller minimizes the overshoot, integral absolute error (IAE) and settling time significantly. The improvement in the performance is due to the optimum parameters of fuzzy system. The quantitative analysis for various controllers is summarized in Table 4, which verifies the effectiveness of the proposed MNFC controller in terms of settling time, overshoot and IAE.

Table 4

Reference tracking performance

Time domain specification	First pendulum angle				Second pendulum angle
	MNFC	GFC	FLC	LQR	MNFC	GFC	FLC	LQR
Settling time (sec)	2.83	3.12	3.32	3.9	3.1	3.36	3.6	4.14
Overshoot (%)	6.95	10.67	11.95	15.1	3.78	6.29	6.91	8.6
Integral absolute error (IAE)	0.1076	0.1431	0.1599	0.1805	0.0689	0.0908	0.1011	0.1138
Steady state error (e_ss)	0	0	0	0	0	0	0	0

4.2 Robustness analysis

Robustness is the ability of the controllers to deal with the perturbations arising in the system due to various factors such as actuator disturbance and sensor noise. The robustness of the designed controllers is examined under the effect of actuator disturbance due to cogging and noise generated by sensor in the feedback path. A quantitative comparison on the basis of IAE value of pendulums angle is made to analyze the performance of controllers.

4.2.1 Disturbance rejection

The robustness of the designed controller is examined by incorporating disturbance to the DIP system. Disturbance can arise in the actuator in oscillatory form due the effect of cogging and eccentricity [35]. Therefore, sinusoidal disturbance (mentioned in Table 5) is added to the input ‘u’ of the system for entire time span. The tracking performance of both the pendulums for MNFC, GFC, FLC and LQR controllers for added disturbance of 2sin25tN is shown in Fig. 13. The values of IAE for 1st and 2nd pendulum under the effect of disturbance are summarized in Table 5. It is observed from the results, that IAE values are smaller for MNFC controller as compared to GFC, FLC and LQR controllers for different cases of disturbance, which is also verified from Fig. 14. Thus, it is observed that proposed MNFC controller rejects the disturbance more efficiently than other designed controllers.

Table 5
IAE values for different cases of disturbance

Added MNFC GFC FLC LQR

Disturbance 1st 2nd 1st 2nd 1st 2nd 1st 2nd

in input (N) pendulum pendulum pendulum pendulum pendulum pendulum pendulum pendulum

1sin25t 0.1159 0.07048 0.1528 0.09246 0.1693 0.1025 0.1898 0.1155

2sin25t 0.1259 0.07249 0.1646 0.09471 0.1794 0.1042 0.1996 0.1172

3sin20t 0.137 0.07488 0.1766 0.09735 0.1893 0.1058 0.2097 0.1189

4sin20t 0.1495 0.07777 0.1856 0.1002 0.199 0.1074 0.2199 0.1206

Added	MNFC	GFC	FLC	LQR
1sin25t	0.1159	0.07048	0.1528	0.09246	0.1693	0.1025	0.1898	0.1155
2sin25t	0.1259	0.07249	0.1646	0.09471	0.1794	0.1042	0.1996	0.1172
3sin20t	0.137	0.07488	0.1766	0.09735	0.1893	0.1058	0.2097	0.1189
4sin20t	0.1495	0.07777	0.1856	0.1002	0.199	0.1074	0.2199	0.1206

Fig.13

Tracking performance of MNFC, GFC, FLC and LQR for disturbance of 2sin25t N (a) 1st pendulum angle (b) 2nd pendulumangle (c) Controller output (d) Disturbance profile.

Fig.14

Variation in IAE for varying amplitude of disturbance (a) 1st pendulum angle (b) 2nd pendulum angle.

4.2.2 Noise suppression

The robustness of the controllers is further investigated by observing their adaptation to the sensor noise in feedback path. A random noise with maximum amplitude of 0.01 radian as shown in Fig. 15(d) is added to the feedback path of 1st pendulum, 2nd pendulum and then both pendulums simultaneously. The tracking performance of designed controllers for addition of random noise to both pendulums is shown in Fig. 15. The values of IAE for each controller after the addition of random noise to the pendulums are recorded in Table 6. A quantitative comparison of IAE value for all controllers is depicted in Fig. 16.

Fig.15

Tracking performance of MNFC, GFC, FLC and LQR for noise added to both pendulums (a) 1st pendulum angle (b) 2nd pendulum angle (c) Controller output (d) Noise profile.

Table 6

IAE Variation for random noise

Added noise in	MNFC		GFC		FLC		LQR
	1st	2nd	1st	2nd	1st	2nd	1st	2nd
	pendulum	pendulum	pendulum	pendulum	pendulum	pendulum	pendulum	pendulum
1st pendulum	0.1226	0.07625	0.1566	0.09828	0.1778	0.1101	0.1965	0.1211
2nd pendulum	0.1365	0.08337	0.1654	0.101	0.1953	0.1185	0.2101	0.1285
Both pendulums	0.1201	0.07539	0.1531	0.09539	0.174	0.1077	0.1941	0.1206

Fig.16

IAE values for different cases of noise (a) 1st pendulum angle (b) 2nd pendulum angle.

MNFC controller has powerful learning capability, so when system is subjected to random noise due to sensor, MNFC tunes the network accordingly and adapts itself to the changes and provides better response by suppressing the noise effectively. The value of IAE for FLC and GFC controllers is comparatively higher, as they are incapable to generalize and take action according to the rule base. Hence, the proposed MNFC controller efficiently suppresses the effect of random noise present in system as compared to other controllers.

4.2.3 Simultaneous disturbance rejection and noise suppression

Effectiveness of the designed controller is also tested for simultaneous inclusion of disturbance and noise to the system. A random noise of amplitude 0.01 radian to feedback path of both pendulums and disturbance of 4sin25tN is added to the input of DIP system for complete time span. The tracking performance and controller output is shown in Fig. 17 and the value of IAE for each controller is recorded in Table 7. The performance of MNFC is again found to be better than other designed controllers.

Fig.17

Tracking performance of MNFC, GFC, FLC and LQR for simultaneous addition of noise and disturbance (a) 1st pendulum angle (b) 2nd pendulum angle (c) Controller output.

Table 7

IAE values for simultaneous addition of disturbance and noise

Added noise and	MNFC		GFC		FLC		LQR
disturbance	1st	2nd	1st	2nd	1st	2nd	1st	2nd
	pendulum	pendulum	pendulum	pendulum	pendulum	pendulum	pendulum	pendulum
Both pendulums	0.1529	0.08133	0.1855	0.1009	0.2025	0.1098	0.222	0.1235
and input

Thus from the rigorous analysis it is concluded that MNFC controller is more efficient and robust than other designed schemes for reference trajectory tracking, disturbance rejection as well as noise suppression.

5 Conclusion

The present work focuses on optimum design of fuzzy logic controller for DIP system with the help of artificial intelligence and evolutionary optimization algorithms. Genetic algorithm based fuzzy controller and adaptive neuro-fuzzy controller are designed by optimizing the FLC parameters using genetic algorithm and neural network respectively. Also a suitable function is proposed for modifying training data set which is utilized in design of neuro-fuzzy inference system that leads to MNFC. Further, FLC and LQR are also designed for comparative analysis. Intensive simulation studies are carried out to test the performance of the controllers for reference tracking, disturbance rejection, noise suppression and simultaneous noise as well as disturbance rejection. The proposed MNFC significantly minimize the IAE as compared to GFC, FLC and LQR for all cases. Thus it is concluded that MNFC exhibits robust and precise control of DIP system as compared to other designed controllers.

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